CAM Centre for Applied Microeconometrics Department of Economics University of Copenhagen http://www.econ.ku.dk/CAM/
Probit Models with Binary Endogenous Regressors
Jacob Nielsen Arendt and Anders Holm
2006-06
The activities of CAM are financed by a grant from The Danish National Research Foundation
Probit models with binary endogenous regressors
Jacob Nielsen Arendta and Anders Holmb
a
Department of Business and Economics, University of Southern Denmark, Odense. email:
[email protected] b
Department of Sociology, University of Copenhagen. email:
[email protected]
February 2006
Abstract Sample selection and endogeneity are frequent causes of biases in non-experimental empirical studies. In binary models a standard solution involves complex multivariate models. A simple approximation has been shown to work well in bivariate models. This paper extends the approximation to a trivariate model. Simulations show that the approximation outperforms full maximum likelihood while a least squares approximation may be severely biased. The methods are used to estimate the influence of trust in the parliament and politicians on votingpropensity. No previous studies have allowed for endogeneity of trust on voting and it is shown to severely affect the results.
Keywords: Endogeneity; Multivariate Probit; Approximation; Monte Carlo Simulation
1. Introduction In this paper we consider how to estimate the effect of endogenous binary variables in a binary response model. This problem is of tremendous importance in most social sciences disciplines, since these frequently rely on non-experimental data. It is well-known from especially linear models that failing to take endogeneity into account may result in substantially biased results. It is natural to assume that such bias extends to non-linear models (Yatchew and Griliches (1985) derive the approximate bias in a probit model with continuous endogenous regressors). We focus on models with qualititative variables for several reasons: 1) Less attention has been paid to these models than when either dependent or independent endogenous variables are continuous, 2) one frequently encounters qualitative variables in social sciences applications, and 3) when both dependent and independent variables are qualitative correct modelling require more complex models than if either is continuous. Specifically, simple two-stage methods exist which account for endogeneity in models when either the dependent or the independent endogenous variable is continuous (see e.g. Alvarez and Glasgow (2000) for a comparison of methods in the latter case), whereas such procedures are generally not consistent with qualitative endogenous variables. Although consistent estimates can be obtained by multivariate modelling, this provides several difficulties both with respect to estimation and with respect to making such models readily understandable to a wider audience. We, as a consequence, think that it is of great value to consider simpler models that approximate the true effects. We consider two types of approximations: a heckit-type and a least-squares type. These are defined below.
For illustration we start by considering a binomial model with one endogenous binomial explanatory variable and present the approximations to the full bivariate model. Nicoletti and
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Perrachi (2001) consider how the heckit-approximation performs in a very similar case; a binomial model with sample selection. The main contribution in this paper is to extend the heckit-approximation to the case with two endogenous binomial explanatory variables and to provide simulation results that illustrate the bias of this approximation as well as of a simpler least-squares-based approximation in different settings. We find that the heckit approximation works well and that it even outperforms full maximum likelihood estimation under serious endogeneity in small samples. The least squares approximation works well under mild endogeneity but may provide seriously biased estimates when endogeneity is severe. To illustrate empirically how the approximation works we apply the heckit approximation for estimation of the effect of political trust on voting behaviour. There is a substantial literature in political sciences on this issue. Nevertheless, according to our knowledge, no previous studies account for the potential endogeneity of trust on voting. We show that taking endogeneity into account has important consequences for the estimated effect of trust on voting.
The paper is organized as follows: The next section presents the case with one endogenous regressor. Section three extends the model to more endogenous or multinomial outcomes. In section four, simulations ecidence for the estimators are presented. Section five presents the empirical application on voting behaviour and section six presents some concluding remarks.
2. A model with one endogenous regression variable In this section we present the case with two binary variables, y1 and y2 , where y2 may have a causal effects on y1 , but where the variables are spuriously related due to observed as well as unobserved independent variables. This situation is illustrated in the following fully parametric model:
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y1 = 1(α y2 + x1β1 + ε1 > 0), (1)
y2 = 1( x2 β 2 + ε 2 > 0), (ε1 , ε 2 | x1 , x2 ) ~ N (0, 0,1,1, ρ ),
where 1(.) is the indicator function taking the value one if the statement in the brackets are true and zero otherwise. α , β1 , β 2 are regression coefficients, N (.,.,.,., ρ ) indicates the standard bivariate normal distribution with correlation coefficients ρ . When ρ is zero the model for y1 is the standard probit model.1
Basically, the model states three reasons why we might observe y1 and y2 to be correlated: 1) a causal relation due to the influence from y2 on y1 through the parameter α , 2) y2 and y1 may depend on correlated observed variables (the x’s) and 3) y2 and y1 may depend on correlated unobserved variables (the ε’s).
Consistent and asymptotically efficient parameter estimates are obtained by maximum likelihood estimation of the bivariate probit model. This is based on a likelihood function consisting of a product of individual contributions of the type: (1) 1
Li (α , β1 , β 2 | yi1 , y2i , x1i , x2i ) = P( yi1 , y2i | x1i , x2i ) = P( yi1 | y2i , x1i ) P( y2i | x2i ).
We stress an important difference between the multivariate probit model and log-linear models. The latter were
considered by Nerlove and Press (1976) and discussed by Heckman (1978) among others. In these models, the bivariate probability of y1 and y2 can be defined as: P ( y1 , y2 ) = exp(α 0 + α1 y1 + α 2 y2 + α12 y1 y2 ) / D , where D is the appropriate weight. In this model, y1 and y2 are independent if and only if α12 is zero. Therefore, it only has one parameter describing the relation between y1 and y2 in contrast to the multivariate probit model which for two types of relations: structural ( α ≠ 0 ) and spurious ( ρ ≠ 0 ), and therefore allows for causal interpretations.
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The second part of the likelihood is simply a probit for y2. The first part of the individual likelihood contributions is given as (see e.g. Wooldridge (2002) p. 478):
P( yi1 = 1| y2i = 1, x1i ) = P(α y2i + x1i β1 + ε1i > 0 | ε 2i > − x2i β 2 ) (2)
∞
=
⎛ α y + x β + ρε ⎞ φ (ε ) 2i 1i 1 2i 2 ⎟ d ε 2i . 2 ⎟ Φ x ( β ) − 1 ρ i 2 2 ⎝ ⎠
∫β Φ ⎜⎜
− x2 i
2
Even though rather precise procedures for evaluation of (2) exist they are often timeconsuming in an iterative optimization context. Furthermore, when ρ approaches one it can be seen from (3) that the integral numerically blows up and estimation becomes imprecise. Both drawbacks are circumvented with an approximation of the following type: (3)
⎛ φ ( x2i β 2 ) ⎞ P(α y2i + x1i β1 + ε1i > 0 | ε 2i > − x2i β 2 ) ≈ Φ ⎜ α y2i + x1i β1 + ρ ⎟. Φ( x2i β 2 ) ⎠ ⎝
The ratio φ /Φ is the inverse Mill’s ratio. Of course, P( yi1 = 0 | y2i = 1, x1i ) can be approximated by one minus this expression. When conditioning on y2i = 0 , a similar approximation holds, replacing φ /Φ by -φ /(1-Φ).
The approximation in (3) is based on the following properties of the normal model:
(4)
E ( y1* | y2* > 0) = α y2i + x1i β1 + ρ E ( ε 2 | ε 2 > − x2i β 2 ) = α y2i + x1i β1 + ρ
φ ( x2i β 2 ) Φ( x2i β 2 )
y1* = α y2i + x1i β1 + ε1i , y2* = x2i β 2 + ε 2i .
Note that the latter pertains to the latent variable y1* . Within the economic literature this is often called the heckit-correction because it was first applied by Heckman (see e.g. Heckman, 1976) in cases where y1* is observed (i.e. when y1 is continuous). Replacing the probabilities given in (3) in the likelihood function by the approximated probabilities, estimates of the parameters of interest can be obtained by the following two-stage procedure: First estimate β 2 in a probit
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model for y2 . Then calculate the correction factors and estimate (α, β1, ρ) in a probit model with the correction factor as additional explanatory variables.
Note that the reason that the correction is an approximation when applied to binomial variables is that it changes mean and indicator functions:
(5)
E ( y1* | y2* > 0) ≠ E (1( y1* > 0) | y2* > 0) = E ( y1 | y2* > 0) = P ( y1 = 1| y2* > 0).
Therefore, this two-stage estimator does not provide consistent estimates, but the approximation of the probability it is based on (that is, (4)) are exact for ρ = 0 (where both are equal to the simple probit), and has been shown to be rather precise for values of ρ even as high as 0.82. Nicoletti and Perrachi (2001) show that the heckit-correction works well in a binomial model with sample selection. This is particularly so if the heteroscedasticity inherent in the correction is taken into account.
Least squares approximation An even simpler alternative approximation than the heckit-correction exists. This would be to use simple least squares residuals as corrections rather than inverse Mill’s ratios. It corresponds to assuming that the qualitative endogenous variable y2 can be modelled linearly as a function of explanatory variables (i.e. with the linear probability model):
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Nicoletti and Perracci (2001) show, via a Taylor approximation of (3) and (4) around ρ = 0 that they are very
close for ρ close to zero and that they are equal for ρ = 0 . They perform a simulation exercise showing that the performance of a similar two-step estimator for a sample selection model is close to that of the bivariate MLE and better than the simple probit for ρ as high as 0.8.
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(6)
y2i = x2i β 2 + ε 2i , P( y1i = 1| y2i ) ≈ Φ (α y2i + x1i β1 + ρε 2i ) .
The linear probability model may often give good estimates of underlying non-linear models. The main problem in this model is that the marginal effect is kept constant. This yields nonsense predictions when the latent variable y2i* is close to zero or one. Indeed, the predicted probability that y2i will be one, x2i β , may be outside the [0,1]-interval. However, in many applications the predicted probabilities are not near the unit-interval boundaries.
3. A Model with two endogenous qualitative regression variables
The heckit approximation presented in the previous section has been shown to perform well (Nicoletti and Peracchi, 2001) in a very similar model. Therefore, we do not consider this model any further. Instead, we explore situations where an approximation may be even more fruitful, namely when the dimension of the endogeneity problem increases. We focus on the case of a binary response model with two endogenous qualitative variables. A simple extension of the model in the previous section that includes one more endogenous discrete regressor would be: y1 = 1(α 2 y2 + α 3 y3 + x1β1 + ε1 > 0), (7)
y2 = 1( x2 β 2 + ε 2 > 0), y3 = 1( x3 β3 + ε 3 > 0), (ε1 , ε 2 , ε 3 | x1 , x2 , x3 ) ~ N (0, 0, 0,1,1,1, ρ12 , ρ13 , ρ 23 ).
Related models arise when we observe y1i under two sample selections restrictions described by y2i and y3i or if we have one qualitative endogenous variable with three unordered outcomes (e.g. z = yes, no, no response. Then, for instance, y2 =1(z=no) and y3 =1(z=no
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response))3. Full maximum likelihood estimation requires estimation of a trivariate probit model, which is consistent and asymptotically efficient. The likelihood function in this case would look like (3), but now with two outer integrals. The trivariate probit estimates can be obtained using numerical integration or simulation techniques. The most common simulation estimator is probably the GHK simulated maximum likelihood estimator of Geweke (1991), Hajivassiliou (1990), and Keane (1994), which is available e.g. in the statistical program packages STATA and LIMDEP (mvprobit, see Cappellari and Jenkins, 2003). However, just as in the bivariate case, we may encounter several practical problems with the trivariate probit model.
Another alternative may again be to use a multivariate heckit-type of approximation: (8)
P(α 2 y2i + α 3 y3i + x1i β1 + ε1i > 0 | ε 2i > − x2i β 2 , ε 3i > − x3i β 2 )
≈ Φ (α 2 y2i + α 3 y3i + x1i β1 + E (ε1i | ε 2i > − x2i β 2 , ε 3i > − x3i β 2 ) ) .
To obtain this approximation we need the first moment in a trivariate truncated normal distribution. It simplifies greatly if we assume ρ 23 = 0 . Then we just get a double heckmancorrection: (9)
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E ( ε1 | ε 2 > − x2i β 2 , ε 3 > − x3i β 3 ) = ρ12
φ ( x3 β 3 ) φ ( x2 β 2 ) + ρ13 . Φ ( x2 β 2 ) Φ ( x3 β 3 )
If the outcomes of ordered, the same correction can still be used. We could make use of the ordering and hence J
increase efficiency by specifying: y2 = ∑ 1(y*2 > c j ) where cj are unobserved thresholds. j=1
The Heckman-correction in this case is: E (ε1 | y2 = j ) = ρ12 E (ε 2 | c j −1 < x2 β 2 + ε 2 < c j ) =
φ ( µ j −1 ) − φ ( µ j ) , µ j = c j − x2 β 2 Φ ( µ j ) − Φ ( µ j −1 )
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In the general case, where ρ 23 ≠ 0 , the correction terms become more complicated. It was applied by Fishe et al. (1981) in a model where y1 is continuous and is found e.g. in Maddala (1983), p. 282:
E ( ε1 | ε 2 < h, ε 3 < k ) = ρ12 M 23 + ρ13 M 32 ; M ij = (1 − ρ 232 )−1 ( Pi − ρ 23 Pj ),
(10)
Pi = E (ε i | ε 2 < h, ε 3 < k ), i = 2,3.
We refer to this as the trivariate heckit correction. Fishe et al. (1981) evaluated these using numerical approximations. They can however be simplified using results found in Maddala (1983), p. 368: P( x > h, y > k ) E ( x | x > h, y > k ) = φ (h) ⎡⎣1 − Φ (k * ) ⎤⎦ + ρφ (k ) ⎡⎣1 − Φ (h* ) ⎤⎦ (11)
h* =
h − ρk 1− ρ
2
, k* =
k − ρh 1− ρ 2
, cov( x, y ) = ρ .
The formulas for the correction terms in the four cases of combinations of Y2 and Y3 being 0 or 1 are presented in the appendix. In order to calculate the two correction terms, M23 and M32, we need initial estimates ofβ2 and β3 and ρ23. This can be obtained from a bivariate probit for Y2 and Y3. Alternatively we may use two linear probability models (LP) for Y2 and Y34. The model therefore involves several steps: 1. Perform estimations for Y2 and Y3 and calculate the correlation between errors (using bivariate probit or linear probability models). 2. Calculate the correction terms 3. Perform a Probit estimation for Y1 adding the correction terms as additional covariates.
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Using the LP estimates of β2 and β3 as initial estimates, we need to rescale them as β2 and β3 estimates are not
from the normal model. The scaling of linear probability coefficients by 2.5 (subtracting 1.25 from the constant) has shown to work well (Maddala, 1983, p. 23). The initial estimate of ρ23 is obtained from the LP model as the correlation between the residuals.
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We have considered whether the performance of the heckit approximation improved if we take into account that it is heteroscedastic. Recall that Nicoletti and Peracchi (2001) found this to be useful in the case with one endogenous regressor. However, as opposed to Nicoletti and Peracchi (2001) we did not find much gain from heteroscedasticity corrections. The formula for the variance needed for heteroscedasticity correction is available from the authors upon request.
Like Nicoletti and Perrachi (2001) we evaluated initially how good an approximation the trivariate heckit gives to the trivariate normal probabilities using graphical illustrations and Taylor-expansions around ( ρ12 , ρ13 ) = (0, 0) . Both the bivariate heckit-correction and the trivariate normal probability have the same first-order Taylor approximations if ρ 23 = 0 : (12)
P (Y1 = 1| Y2 = 0, Y3 = 0) ≈ Φ ( x1β1 ) + ρ12
φ ( x3 β3 ) φ ( x2 β 2 ) φ ( x1β1 ) + ρ13 φ ( x1β1 ). Φ ( x2 β 2 ) Φ ( x3 β 3 )
The trivariate heckit (with ρ 23 ≠ 0 ) has a similar first-order Taylor-expansion where one replaces the inverse of the Mill’s ratios by M23 and M32 found in (10) and (11). By simulation it is found that especially the trivariate heckit approximates the true normal probability rather well even for high correlation coefficients, whereas the bivariate heckit is often badly behaved (when ρ 23 ≠ 0 ). However, over some ranges of outcomes and with some correlation coefficients the approximation of the trivariate heckit also performs poorly. The Taylor expansion as well as graphical illustrations are found in the appendix.
4. Simulations
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In this section we report simulation results demonstrating the performance of the trivariate heckit-approximation and OLS-based approximation both described above. The model consists of the following three endogenous variables:
(13)
y1* y2*
= β10 + β11 x11 + β12 x12 + γ 1 y2 + γ 2 y3 + e1 , = β 20 + β 21 x21 + β 22 x22 + e2 ,
y3*
= β 30 + β 31 x31 + β 32 x32 + e3 .
with ( β j 0 , β j1 , β j 2 ) = (0, 0.5, −0.5), j = 1, 2, 3, (γ 1 , γ 2 ) = (0.5, 0.5) and:
(14)
⎛ ⎡ 0 ⎤ ⎡ 1 σ 12 σ 13 ⎤ ⎞ ⎡ e1 ⎤ ⎢ e ⎥ ∼ N ⎜ ⎢ 0 ⎥ , ⎢σ ⎥⎟ ⎜ ⎢ ⎥ ⎢ 12 1 σ 23 ⎥ ⎟ , ⎢ 2⎥ ⎜ ⎢ 0 ⎥ ⎢σ ⎟ ⎢⎣ e3 ⎥⎦ ⎝ ⎣ ⎦ ⎣ 13 σ 23 1 ⎥⎦ ⎠
⎧1 if y*j ≥ 0 yj = ⎨ , j = 1, 2,3, ⎩0 otherwise
where y2 and y3 are endogenous to y1 if the e2 or e3 are correlated with e1. Furthermore, if e2 and e3 are correlated a full trivariate model is required. This is where the trivariate heckit approximation is expected to be most relevant. The regressors are drawn as independent standard normal variables with 500 independent draws in each simulation. The tri-variate heckit is based on initial scaled (see footnote 3) OLS estimates of parameters in the equations for y2 and y3 and so is the estimate of σ 23 . For comparison we have also simulated the trivariate probit using the GHK simulated MLE5. We apply the rule-of-thumb (see Cappellari and Jenkins, 2003) that the number of draws made by the GHK estimator for each simulation is the square root of the number of observations, here 23. Experimenting with the number of simulations shows that results do not change when altering the number of Monte Carlo simulations from 200 to 1000, hence 200 is used.
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This is done using STATA vrs. 9.0 and the mvprobit procedure written by Cappellari and Jenkins. The other
simulations are conducted in GAUSS. The trivariate heckit correction is available upon request from the authors in both GAUSS and STATA code.
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In table 1a-c we show average estimation results with various combinations of values of correlations between the three error terms. In table 1a we show results when all correlations are zero. To save space, we mainly comment on results for effects of y2 and y3 in the y1 equation and only these estimates along with the correlation coefficients. Table 1a. Simulation results with no endogenity. True Single equation Bi-variate Heckit probit Bias MSE Bias MSE 0.5 γ1 0.015 0.015 0.024 0.105 0.5 γ2 0.009 0.015 0.015 0.113 0.0 σ 12 -0.005 0.049 0.0 σ 13 0.0009 0.050 0.0 σ 23
Tri-variate Heckit
OLS residuals
Bias 0.024 0.016 -0.006 0.0005
Bias
MSE 0.104 0.113 0.048 0.050
Trivariate Probit
MSE
Bias
0.025 0.076 -0.018 0.017 0.084 0.001 -0.010 0.097 0.015 -0.003 0.103 0.009 -0.004 0.002 0.010 Note: Each simulation consists of 500 draws from the data generating process. Estimates and standard deviations reported in the table are based on 200 simulations.
From the table we find that all estimators are unbiased. The multivariate methods however have a larger mean squared error (MSE) than the simple probit estimator. The MSE of the multivariate approaches are reasonably close, OLS and trivariate probit performing slightly better than the two heckit approaches which show similar performance. The higher MSE is the cost of using a multivariate procedure when it is not needed. The gain of is that one has an assessment of the degree of endogeneity in form of estimates of σ 12 and σ 13 and the test that these parameters are zero is test of exogeneity of y2 and y3 . As one can see the estimated correlations are fairly close to zero, although the approximations are somewhat less precise than the trivariate probit.
In table 1b we show results when there is a moderate degree of endogenity as well as correlation between the two endogenous explanatory variables.
Table 1b. Simulation results with moderate degree of endogenity and correlation
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MSE 0.079 0.075 0.031 0.005 0.035
between endogenous explanatory variables. True Single equation Bi-variate Heckit Tri-variate Heckit probit Bias MSE Bias MSE Bias MSE 0.5 -0.082 0.113 γ1 0.355 0.140 -0.081 0.113 0.5 -0.070 0.135 γ2 0.354 0.145 -0.071 0.134 0.3 0.054 0.056 σ 12 0.006 0.053 0.3 0.047 0.067 σ 13 -0.002 0.064 σ 23 0.5 Note: See table 1a. 3 of the trivariate probit estimations failed to converge.
OLS residuals
Trivariate Probit
Bias
Bias
0.006 0.023 0.123 0.098 -0.230
MSE 0.070 0.086 0.111 0.123 0.055
0.068 0.054 -0.030 -0.032 0.050
MSE 0.081 0.071 0.041 0.036 0.011
From the table we see that the single equation probit that completely ignores endogeneity has a large positive bias. From the table it is also clear that even though the approximations are far better than the single equation probit, they are not able to completely recover the true values. But neither is the trivariate probit. In fact, the trivariate probit does not outperform the OLS correction neither in terms of bias or MSE. Moreover, the two heckit approximations have only a slightly higher bias and MSE than trivariate probit.
In table 1c we show simulation results when there is a high correlation between the error terms in the two equations for y2 and y3 in addition to a high correlation between the error terms in the equations of y2 and y3 and the error term in the equation for y1.
Table 1c. Simulation results with strong degree of endogenity and and strong positive correlation between endogenous explanatory variables. True Single equation Bi-variate Heckit Tri-variate Heckit OLS residuals Trivariate Probit probit Bias MSE Bias MSE Bias MSE Bias MSE Bias MSE 0.5 -0.051 0.039 γ1 1.017 1.056 -0.180 0.077 0.282 0.102 0.149 0.099 0.5 -0.070 0.036 γ2 0.999 1.022 -0.199 0.076 0.263 0.093 0.167 0.088 0.75 0.221 0.055 σ 12 0.203 0.052 0.239 0.059 0.098 0.062 0.75 0.221 0.056 σ 13 0.203 0.053 0.238 0.060 0.080 0.039 0.50 σ 23 -0.230 0.055 0.061 0.011 Note: See table 1a. 4 of the trivariate probit estimations failed to converge.
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From the table we again find a large bias for the two endogenous explanatory variables in the single equation probit model. The bias of the trivariate heckit is relatively low whereas all the three other multivariate methods have a non-negligible bias, including the trivariate probit and the OLS-correction. The trivariate heckit also outperforms the other estimators in terms of MSE. The correlation coefficients are however biased for the approximations while far closer to the true values for the trivariate probit.
We have made simulations for a model with very similar true values as in table 1c, except that the correlation of the error term in the equation of y3 and the two other error terms are negative. The findings from this exercise are similar to findings in table 1c. The caveat in this simulation is that under certain parameterizations the probit does not even get the sign right for the coefficient for y3.
Finally, it is worth noting that in all simulations the estimated coefficients for the exogenous explanatory variables (except the constant term) seem to be well-estimated in the single equation probit model, irrespective of the severity of the endogenity of y2 and y3. This is surprising and may be due to the fact that all regressors are assumed different and uncorrelated.
An application of voting and trust
In this section we use an empirical application to illustrate how endogeneity of binomial indicators in binomial models may affect the estimated effects. The empirical example is a study of the effect of trust on voting behaviour.
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There are several examples in the literature seeking to estimate the effect of trust on voting behaviour. In Pattie and Johnston (2001), voting in the 1997 election in the UK is analysed using both trust indices and previous voting behaviour as explanatory variables. In Peterson and Wrighton (1998) voting at the four previous US presidential elections is analysed also using trust, through the trust in government index from Miller (1974), on voting behaviour at the US presidential elections. Cox (2003) analyses voter turn out at European parliament elections using a variety of trust measures. All these studies treat trust as exogenous. However, one can imagine several reasons why this assumption may fail.
First of all, since voting behaviour is often reported as voting in the latest election (which is also the case in our application), there might be a problem of reverse causality. Information obtained since the last election about how the current politicians and parliament have performed, might affect the responses on trust in politicians and the parliament. Second, trust is a subjective measure, and might thus be contaminated by substantial measurement error, which also make trust an endogenous variable. Finally, spurious relations (unobserved heterogeneity) might in general make trust variables endogenous. For example, people who have a general positive attitude are more likely to vote as well as being more likely to trust other, leaving attitude out of the model will induce a spurious relationship between voting and trust. In some studies on voting behaviour the trust variables are viewed as indicators of social capital (e.g. Cox, 2003). If social capital is the reason why trust and voting are related, it is likely that social capital is not fully described by trust, and hence a host of other indicators may be correlated with voting behaviour. However, if other dimensions of social capital relevant to voting behaviour, while not being included in the model, are also related to trust they are swept into the error term and will induce endogeneity of the trust variables.
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None of the mentioned studies acknowledge the potential endogeneity of the trust variables. An exception is the related case studied by Alvarez and Glasgow (2000), who consider how voter uncertainty on the political candidate’s policy position affects voting behaviour. They take endogeneity into account but use a continuous measure of voter uncertainty, thus they consider another class of estimators than the ones described here.
In our application we show that endogenity is a serious problem and whether it is taken into account or not has serious implications for the results obtained. To encompass our application to the methods of several binary endogenous variables, we use two trust variables; namely trust in politicians and trust in the parliament.
In our exampled the response variable (y1 in (1)) is whether the respondent voted in the last national election. The endogenous variable (y2 in (1)) is whether the respondent has trust in the national parliament. Data comes from the European social survey (ESS), see http://www.europeansocialsurvey.org/ for further documentation on the data. We have sampled 3,651 randomly cases among eligible voters only in all countries in the ESS. However, we exclude a country indicator as, in preliminary analysis, it turned out that although significant on vote, the exclusion of this variable did not affect the estimate of trust on voting behaviour, and hence we feel justified to leave it out of the model for simplicity.
In table 2, we show summary statistics for our sample.
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Table 2. Summary statistics. Variable Mean Vote (yes = 1/no = 0) 0.83 Trust in parliament (yes/no) 0.57 Trust in politicians (yes/no) 0.44 Age 47.99 Gender (female = 1/male = 0) 0.52 Years of education 11.92 Number of observations = 3651
Standard deviation 17.15 3.96
Min 0 0 0 18 0 0
Max 1 1 1 102 1 25
In table 3 we show estimation results for a univariate probit, the tri-heckit estimator presented in this paper and a full information maximum likelihood estimator, the trivariate probit. From the simulations we concluded that the tri-variate probit was not consistently better than the trivariate heckit estimator but much more cumbersome from a computational perspective.
Table 3. Estimations results for voting. Model Univariate probit Eq. for vote Coefficient St error Constant -1.451 0.240 Trust in parliament 0.355 0.061 Trust in politicians 0.121 0.063 Age/10 0.631 0.079 Age squared/100 -0.050 0.008 Years of educ./10 0.312 0.074 Female 0.064 0.054 Eq. for trust in parliament* Constant Age/10 Age squared/100 Years of educ./10 Years of educ. sq./100 Female Eq. for trust in politicians* Constant Age/10 Age squared/100 Years of educ./10 Years of educ. sq./100 Female ρ12
ρ13 ρ 23
-
Tri-heckit Coefficient St error -2.018 0.238 2.343 0.475 -1.027 1.377 0.582 0.082 -0.045 0.008 0.253 0.175 0.113 0.062
-
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Trivariate probit Coefficient St error -1.168 0.193 0.757 0.115 -0.629 0.172 .0557 0.007 -4E-04 7E-5 0.033 0.007 0.090 0.046
-0.113 0.125 -0.009 -0.378
0.008 6.9E-4 6.6E-06 0.008
-0.185 0.0156 -1E-04 -0.057
0.212 0.007 7E-05 0.023
0.300 -0.030
0.001 2.6E-4
0.004 -0.030
0.001 0.042
-0.993 0.086 -5.9E-4 0.113
0.008 7.0E-4 6.7E-06 0.008
-0.950 0.010 -2E-05 0.013
0.218 0.007 7E-05 0.024
0.131 0.061
0.001 2.7E-4
0.001 -2.7E-4
0.001 0.041
-1.000 0.289 0.540
0.440 0.780 -
-0.048 0.277 0.778
0.056 0.084 0.014
Note: *For the Tri-heckit model, these are the re-scaled OLS regression coefficients and the estimate of ρ 23 is the correlation between the OLS residuals. Based on 3651 observations.
From the table we find that both trust in the parliament and trust in the politicians increases the likelihood of voting in the univariate probit. However, both the trivariate heckit as well as the trivariate probit agrees that only trust in the parliament increases the likelihood of voting, whereas trust in the politicians decreases the likelihood of voting. Hence, it appears that the univariate probit completely misses the qualitative relationship between trust in politicians and voting. It appears from all models, with varying effect and significance, that if the electorate trusts the institutional set up of representative democracy they are more likely to vote. This makes sense: If you believe in the system, you are more likely to use it. However, the univariate probit completely disagrees with the two other models on the impact of trust in politicians. But the negative relationship between trust in politicians and voting, predicted by both the tri-variate heckit as well as the trivarite probit, also makes sense: If you have trust in the politicians you are likely to gain less from voting than if you do not trust them. Therefore, if you do not trust politicians you have a higher incentive to vote in order to change the composition of the parliament.
From the table we find evidence of spurious correlation between trust and voting: the error terms between voting and trust in parliament (ρ12) are negatively correlated (only significant in the tri-heckit) and the error terms between trust in the politicians and voting (ρ13) are positively correlated (only significant in the trivariate probit). Finally, the error terms between trust in the parliament and politicians are positively correlated (ρ23).
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Ignoring these correlations, as in the single equation probit, implies that trust in parliament and trust in politicians captures both the causal effect of the trust indicators on voting as well as a spurious effect between voting and trust. For trust in politicians it turns out that the positive spurious relation outweighs the negative causal effect, producing a positive estimate in the simple probit model. For trust in the parliament the negative spurious relation with voting implies that the single equation probit model greatly underestimate the causal effect of trust in the parliament.
5. Conclusion
We have introduced an approximation of a binomial normal model with two binomial endogenous regressors as an alternative to the more complex trivariate probit model. We considered the small sample properties of the approximation and of a simple OLS-based approximation. We showed that a standard probit model that does not account for endogeneity is severely biased in the presence of even moderate endogeneity. The approximations are less biased. This is particularly so for the heckit approximation when the degree of endogeneity is severe. In the latter case, the bias of both the OLS-based approximation and the trivariate probit are not neglible and the efficiency loss of both approximations compared to the standard probit is small. From our application we show both the importance of taking into account endogeneity of binary variables and that the trivarite heckit estimator is a useful tool for doing so. When ignoring endogenity one gets very different estimates compares to what is obtained from models that corrects for endogenity. In certain cases one even gets different signs of the effects of the endogenous variables.
Acknowledgements
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We would like to thank Mads Meier Jæger and participants at the 28th Symposium for Applied Statistics in Copenhagen, 2006, for useful comments.
References
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Appendix 1. The correction terms in the trivariate heckit
The formulas needed for the trivariate Heckman correction are derived. Two formulas from Maddala (1983) are used repeatedly. They are: (*)
E ( ε1 | ε 2 < h, ε 3 < k ) = ρ12 M 23 + ρ13 M 32 2 −1 M ij = (1 − ρ 23 ) ( Pi − ρ 23 Pj ), Pi = E (ε i | ε 2 < h, ε 3 < k ), i = 2,3
and Rosenbaum’s formula (Rosenbaum, 1961): Y2 = 0,1 and Y3 = 0,1 P( x > h, y > k ) E ( x | x > h, y > k ) = φ (h) ⎡⎣1 − Φ (k * ) ⎤⎦ + ρφ (k ) ⎡⎣1 − Φ (h* ) ⎤⎦ (**)
h* =
h − ρk 1− ρ
2
, k* =
k − ρh 1− ρ 2
, cov( x, y ) = ρ
There are four terms in the correction corresponding to pairs of combinations of. To derive these, the following change of variables is used: z = −ε 2 , v = −ε 3 . This is simple, since the transformations have Jacobian equal to one. Note also that: cov(ε1 , z ) = − ρ12 , cov(ε1 , v) = − ρ13 , cov( z , v) = ρ 23 Starting with the first: E (ε1 | Y2 = 1, Y3 = 1) = E (ε1 | ε 2 > − x2 β 2 , ε 3 > − x3 β3 ) = E (ε1 | z < x2 β 2 , v < x3 β 3 )
22
Now we can use (*) to get: E (ε1 | z < x2 β 2 , v < x3 β 3 ) = − ρ12 M 23 − ρ13 M 32 M ij = (1 − ρ 232 ) −1 ( Pi − ρ 23 Pj ) P2 = E ( z | z < x2 β 2 , v < x3 β3 ), P3 = E (v | z < x2 β 2 , v < x3 β3 ) In order to obtain the latter parts, we need to rearrange Rosenbaum’s formula, (**). This is done using the same change of variables as above: P( x < h, y < k ) E ( x | x < h, y < k ) = − P( z > − h, v > − k ) E ( z | z > − h, v > − k )
Note that since the mean is taken of x, which changes sign when changing variables, we get a minus in front of the entire expression. Inserting this in (**) gives: ⎡ ⎡ −k + ρ h ⎤ −h + ρ k ⎤ ) ⎥ − ρφ (− k ) ⎢1 − Φ ( )⎥ −φ (− h) ⎢1 − Φ ( 1 − ρ 2 ⎥⎦ 1 − ρ 2 ⎥⎦ ⎢⎣ ⎢⎣ E ( x | x < h, y < k ) = Φ ( x < h, y < k ; ρ ) ρ = co rr ( x, y )
Therefore: P2 = E ( z | z < x2 β 2 , v < x3 β 3 ) ⎡ ⎡ − x β + ρ 23 x2 β 2 ⎤ − x β + ρ 23 x3 β 3 ⎤ ) ⎥ − ρ 23φ (− x3 β3 ) ⎢1 − Φ ( 2 2 )⎥ −φ (− x2 β 2 ) ⎢1 − Φ ( 3 3 ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ 1 − ρ 232 1 − ρ 232 = Φ( x2 β 2 , x3 β 3 ; ρ 23 ) and P3 is obtained by interchanging x2 β 2 and x3 β 3 . Proceeding in the same fashion, we get: E (ε1 | Y2 = 0, Y3 = 1) = E (ε1 | ε 2 < − x2 β 2 , v < x3 β 3 ) = ρ12 M 23 − ρ13 M 32 M ij = (1 − ρ 232 ) −1 ( Pi + ρ 23 Pj ) P2 = E (ε 2 | ε 2 < − x2 β 2 , v < x3 β3 ), P3 = E (v | ε 2 < − x2 β 2 , v < x3 β 3 )
Note that ρ 23 has also changed sign since it is the correlation between ε 2 and v . Again the adjusted Rosenbaum-formula gives us:
23
P2 = E (ε 2 | ε 2 < − x2 β 2 , v < x3 β 3 ) = ⎡ ⎡ − x β + ρ 23 x2 β 2 ⎤ x β − ρ 23 x3 β 3 ⎤ −φ ( x2 β 2 ) ⎢1 − Φ ( 3 3 ) ⎥ + ρ 23φ (− x3 β 3 ) ⎢1 − Φ ( 2 2 )⎥ ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ 1 − ρ 232 1 − ρ 232 Φ (− x2 β 2 , x3 β3 ; − ρ 23 ) P3 = E (v | ε 2 < − x2 β 2 , v < x3 β 3 ) = ⎡ ⎡ − x β + ρ 23 x2 β 2 ⎤ x β − ρ 23 x3 β3 ⎤ −φ (− x3 β 3 ) ⎢1 − Φ ( 2 2 )⎥ ) ⎥ + ρ 23φ ( x2 β 2 ) ⎢1 − Φ ( 3 3 2 ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ 1 − ρ 23 1 − ρ 232 Φ (− x2 β 2 , x3 β3 ; − ρ 23 )
The third correction is: E (ε1 | Y2 = 1, Y3 = 0) = E (ε1 | z < x2 β 2 , ε 3 < − x3 β 3 ) = − ρ12 M 23 + ρ13 M 32 M ij = (1 − ρ 232 ) −1 ( Pi + ρ 23 Pj ) P2 = E ( z | z < x2 β 2 , ε 3 < − x3 β 3 ), P3 = E (ε 3 | z < x2 β 2 , ε 3 < − x3 β3 )
The adjusted Rosenbaum-formulas are now: P2 = E ( z | z < x2 β 2 , ε 3 < − x3 β 3 ) = ⎡ ⎡ − x β + ρ 23 x3 β 3 ⎤ x β − ρ 23 x2 β 2 ⎤ ) ⎥ + ρ 23φ ( x3 β 3 ) ⎢1 − Φ ( 2 2 )⎥ −φ (− x2 β 2 ) ⎢1 − Φ ( 3 3 2 ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ 1 − ρ 232 1 − ρ 23 Φ (− x2 β 2 , x3 β3 ; − ρ 23 ) P3 = E (ε 3 | z < x2 β 2 , ε 3 < − x3 β 3 ) = ⎡ ⎡ x β − ρ 23 x2 β 2 ⎤ − x β + ρ 23 x3 β3 ⎤ )⎥ ) ⎥ + ρ 23φ (− x2 β 2 ) ⎢1 − Φ ( 3 3 −φ ( x3 β 3 ) ⎢1 − Φ ( 2 2 ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ 1 − ρ 232 1 − ρ 232 Φ (− x2 β 2 , x3 β3 ; − ρ 23 )
Finally the final correction terms are: E (ε1 | Y2 = 0, Y3 = 0) = E (ε1 | ε 2 < − x2 β 2 , ε 3 < − x3 β 3 ) = ρ12 M 23 + ρ13 M 32 M ij = (1 − ρ 232 ) −1 ( Pi − ρ 23 Pj ) Pi = E (ε i | ε 2 < − x2 β 2 , ε 3 < − x3 β 3 ), i = 2,3
where:
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Pi = E (ε i | ε 2 < − x2 β 2 , ε 3 < − x3 β3 ) = ⎡ ⎡ x β − ρ 23 x2 β 2 ⎤ x β − ρ 23 x3 β3 ⎤ ) ⎥ − ρ 23φ ( x3 β 3 ) ⎢1 − Φ ( 2 2 )⎥ −φ ( x2 β 2 ) ⎢1 − Φ ( 3 3 2 ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ 1 − ρ 23 1 − ρ 232 Φ(− x2 β 2 , − x3 β 3 ; ρ 23 )
Appendix 2. Taylor expansions
We show that the bivariate heckit correction has the same first order Taylor expansion around ( ρ12 , ρ13 ) = (0, 0) as the trivariate conditional probability P(Y1=1| Y2=0,Y3=0) of the multivariate probit under the assumption that ρ 23 = 0 . We also derive the 1.order Taylor expansion of the trivariate heckit.
Starting with the latter:
P(Y1 = 1| Y2 = 0, Y3 = 0) ≈ Φ ( x1β1 + ρ12 M 23 + ρ13 M 32 ) ≈ Φ( x1β1 ) + ρ12
∂Φ ( x1β1 + ρ12 M 23 + ρ13 M 32 ) ∂Φ ( x1β1 + ρ12 M 23 + ρ13 M 32 ) |( ρ12 , ρ13 ) =(0,0) + ρ13 |( ρ12 , ρ13 ) =(0,0) ∂ρ12 ∂ρ13
= Φ( x1β1 ) + ρ12 M 23φ ( x1β1 ) + ρ13 M 32φ ( x1β1 ) It is clear that if ρ 23 = 0 (i.e. for the bivariate heckit correction), the same formula is obtained with the M-functions replaced by the standard inverse Mill’s ratios: P (Y1 = 1| Y2 = 0, Y3 = 0) ≈ Φ ( x1β1 + ρ12 λ2 + ρ13λ3 ) ≈ Φ ( x1β1 ) + ρ12 λ ( x2 β 2 )φ ( x1β1 ) + ρ13λ ( x3 β 3 )φ ( x1β1 ) Next we look at the trivariate multivariate probit probabilities. For simplicity we have only found the 1.order Taylor expansion under the assumption that ρ 23 = 0 . Using Bayes’ formula:
25
P (Y1 , Y2 , Y3 ) = P (Y2 , Y3 | Y1 ) P(Y1 ) = P(Y2 | Y1 ) P (Y3 | Y1 ) P(Y1 ) = P(Y1 , Y2 ) P(Y1 , Y3 ) / P(Y1 ) i.e. P (Y1 = 1| Y2 = 0, Y3 = 0) = P (Y1 = 1, Y2 = 0) P (Y1 = 0, Y3 = 0) /( P (Y1 = 1) P(Y2 = 0) P(Y3 = 0))
With the latent variable structure we can write these probabilities in the usual way with the normal cdf evaluated at appropriate indices, which we for simplicity denotes by a’s here. The Taylor-expansion of this is: P (Y1 = 1| Y2 = 0, Y3 = 0) = Φ (a1 , a2 )Φ (a1 , a3 ) /(Φ (a1 )Φ (a2 )Φ (a3 )) ≈ Φ (a1 )Φ (a2 )Φ (a1 )Φ ( a3 ) /(Φ ( a1 )Φ (a2 )Φ (a3 )) + ρ12 [ + ρ13 [
∂Φ (a1 , a2 ) Φ (a1 , a3 ) /(Φ (a1 )Φ (a2 )Φ (a3 ))]( ρ12 , ρ13 ) =(0,0) ∂ρ12
∂Φ (a1 , a3 ) Φ (a2 , a3 ) /(Φ (a1 )Φ (a2 )Φ (a3 ))]( ρ12 , ρ13 ) =(0,0) ∂ρ13
Noting that the derivative of the bivariate distribution function with respect to the correlation is just the bivariate density, we get: P (Y1 = 1| Y2 = 0, Y3 = 0) ≈ Φ (a1 ) + ρ12
φ (a1 )φ (a2 ) Φ ( a2 )
+ ρ13
φ (a1 )φ (a3 ) Φ (a1 )
= Φ (a1 ) + ρ12φ (a1 )λ (a2 ) + ρ13φ (a3 )λ (a1 )
i.e. the same as for the bivariate heckit. We have plotted the Taylor-approximation along with the true trivariate probabilities and the bivariate and trivariate heckit approximations. These are shown in three cases in figure 2-4. In most scenarios do the trivariate heckit work well, but in many cases this is not the case for the bivariate heckit. Figure 2 shows a case where all are alike. Figure 3 shows a case where the bivariate heckit does not work well whereas the trivariate does (since r23 is not zero), and finally figure 4 shows a case where the trivariate heckit does not work well. It is worth noticing that the Taylor expansions often work better than both of the heckit methods. But of coursem this is only selective evidence.
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Figure 2. Example where the heckit approximations work well:
Figure 3. Example where trivariate heckit works, but the bivariate heckit does not work well:
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Figure 4. Example where the trivariate heckit does not work well:
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